Q. 36 I5.0( 1 Vote )

Find one-parameter families of solution curves of the following differential equations:

Answer :

Given

This is a first order linear differential equation of the form

Here, P = sec2x and Q = tan x sec2x

The integrating factor (I.F) of this differential equation is,

We have

I.F = etan x

Hence, the solution of the differential equation is,

Let tan x = t

sec2xdx = dt [Differentiating both sides]

By substituting this in the above integral, we get

Recall

yet = tet – et + c

yet × e–t = (tet – et + c)e–t

y = t – 1 + ce–t

y = tan x – 1 + ce–tan x [ t = tan x]

Thus, the solution of the given differential equation is y = tan x – 1 + ce–tan x

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